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Mitze, Timo
Working Paper
Endogeneity in Panel Data Models with Time-Varying
and Time-Fixed Regressors: To IV or not IV?
Ruhr economic papers, No. 83
Provided in Cooperation with:
RWI - Leibniz-Institut für Wirtschaftsforschung, Essen
Suggested Citation: Mitze, Timo (2009) : Endogeneity in Panel Data Models with TimeVarying and Time-Fixed Regressors: To IV or not IV?, Ruhr economic papers, No. 83, ISBN
978-3-86788-093-0
This Version is available at:
http://hdl.handle.net/10419/26848
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#83
Ruhr Economic Papers
Timo Mitze
Ruhr Economic Papers
Published by
Ruhr-Universität Bochum (RUB), Department of Economics
Universitätsstraße 150, 44801 Bochum, Germany
Technische Universität Dortmund, Department of Economic and Social Sciences
Vogelpothsweg 87, 44227 Dortmund, Germany
Universität Duisburg-Essen, Department of Economics
Universitätsstraße 12, 45117 Essen, Germany
Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI Essen)
Hohenzollernstrasse 1/3, 45128 Essen, Germany
Editors:
Prof. Dr. Thomas K. Bauer
RUB, Department of Economics
Empirical Economics
Phone: +49 (0) 234/3 22 83 41, e-mail: [email protected]
Prof. Dr. Wolfgang Leininger
Technische Universität Dortmund, Department of Economic and Social Sciences
Economics – Microeconomics
Phone: +49 (0) 231 /7 55-32 97, email: [email protected]
Prof. Dr. Volker Clausen
University of Duisburg-Essen, Department of Economics
International Economics
Phone: +49 (0) 201/1 83-36 55, e-mail: [email protected]
Prof. Dr. Christoph M. Schmidt
RWI Essen
Phone: +49 (0) 201/81 49-227, e-mail: [email protected]
Editorial Office:
Joachim Schmidt
RWI Essen, Phone: +49 (0) 201/81 49-292, e-mail: [email protected]
Ruhr Economic Papers #83
Responsible Editor: Thomas K. Bauer
All rights reserved. Bochum, Dortmund, Duisburg, Essen, Germany, 2009
ISSN 1864-4872 (online) – ISBN 978-3-86788-093-0
The working papers published in the Series constitute work in progress circulated to
stimulate discussion and critical comments. Views expressed represent exclusively
the authors’ own opinions and do not necessarily reflect those of the editors.
Ruhr Economic Papers
#83
Timo Mitze
Bibliografische Information der Deutschen Nationalbibliothek
Die Deutsche Nationalbibliothek verzeichnet diese Publikation in
der Deutschen Nationalbibliografie; detaillierte bibliografische Daten
sind im Internet über http://dnb.d-nb.de abrufbar.
ISSN 1864-4872 (online)
ISBN 978-3-86788-093-0
Timo Mitze*
Endogeneity in Panel Data Models with Time-Varying and
Time-Fixed Regressors: To IV or not IV?
Abstract
We analyse the problem of parameter inconsistency in panel data econometrics due to the correlation of exogenous variables with the error term. A common solution in this setting is to use Instrumental-Variable (IV) estimation in
the spirit of Hausman-Taylor (1981). However, some potential shortcomings
of the latter approach recently gave rise to the use of non-IV two-step estimators. Given their growing number of empirical applications, we aim to systematically compare the performance of IV and non-IV approaches in the presence of time-fixed variables and right hand side endogeneity using Monte
Carlo simulations, where we explicitly control for the problem of IV selection
in the Hausman-Taylor case. The simulation results show that the HausmanTaylor model with perfect-knowledge about the underlying data structure (instrument orthogonality) has on average the smallest bias. However, compared
to the empirically relevant specification with imperfect-knowledge and instruments chosen by statistical criteria, the non-IV rival performs equally well or
even better especially in terms of estimating variable coefficients for timefixed regressors. Moreover, the non-IV method tends to have a smaller root
mean square error (rmse) than both Hausman-Taylor models with perfect and
imperfect knowledge about the underlying correlation between r.h.s variables
and residual term. This indicates that it is generally more efficient. The results
are roughly robust for various combinations in the time and cross-section dimension of the data.
JEL Classification: C15, C23, C52
Keywords: Endogeneity, instrumental variables, two-step estimators, Monte
Carlo simulations
January 2009
* Ruhr-University Bochum & Gesellschaft für Finanz- und Regionalanalysen (GEFRA)
Münster. – The author thanks Jan Jacobs and further participants of the NAKE Research Day
2008 at Utrecht University for helpful comments and advices. – All correspondence to Timo
Mitze, e-mail: [email protected].
1
Introduction
In contemporary panel data analysis researchers are often confronted with the problem
of parameter inconsistency due to the correlation of some of the exogenous variables with
the model’s error term. Assuming that this correlation is typically due to unobservable
individual effects (see e.g. Mundlak, 1978), a consistent approach to deal with such type
of right hand side endogeneity is to apply the standard Fixed Effects Model (FEM), which
uses a within-type data transformation to erase the unobserved individual effects from
the model. However, one drawback of this estimator is that the within transformation
also wipes out all explanatory variables that to not change in the time dimension of the
model. In this case no statistical inference can be made for these variables, if they have
been included in the original untransformed model based on theoretical grounds. Likewise,
the Random Effect Model (REM), which rests upon the strong assumption of exogeneity
of all right hand side regressors with respect to the error term, is biased for the case when
endogeneity occurs.
The researcher’s problem is then to find a consistent estimator, which is still capable
of including time-fixed regressors in the estimation setup. A well-known example for the
above sketched etimation setup in empirical work is the gravity model (of trade, capital or
migration flows among other interaction effects), which assigns a prominent role given to
time-fixed variables in the regression model.1 In this paper we thus aim to focus on proper
estimation strategies for Gravity type and related models, when some time-varying and
-fixed right hand side regressors are correlated with the unobservable individual effects.
Baltagi et al. (2003) have shown, that when there is endogeneity among the right hand side
regressors the OLS and Random Effects estimators are substantially biased and both yield
misleading inference. As an alternative solution the Hausman-Taylor (1981, thereafter
HT) approach is typically applied. The HT estimator allows for a proper handling of data
settings, where some of the the regressors are correlated with the individual effects. The
estimation strategy is basically based on IV methods, where instruments are derived from
internal data transformations of the variables in the model, thus no external information
for model estimation is necessary. One of the advantages of the HT model is that it
avoids the ’all or nothing’ assumption with respect to the correlation between right hand
1 Taking the gravity model of trade as an example, the model is a beloved playground for applied econometric work:
With the recent switch from cross-section to panel data specifications, important shortcomings of earlier gravity model
applications have been tackled (see e.g. Matyas, 1997, Breuss & Egger, 1999, as well as Egger, 2000), however, other
methodological aspects such as the proper functional form of the Gravity equation are still subject to open debate in the
recent literature (see e.g. Baldwin & Taglioni, 2006, and Henderson & Millimet, 2008, for an overview). Recently, also the
time series properties of Gravity models have reached the center of academic research (see e.g. Fidrmuc, 2008, Zwinkels &
Beugelsdijk, 2008).
4
side regressors and error components, which is made in the standard FEM and REM
approaches respectively. However, for the HT model to be operable, the researcher needs
to classify variables as being correlated and uncorrelated with the individual effects, which
is often not a trivial task.
As a response of this drawback in empirical application of the HT approach different
estimation strategies have been suggested, which strongly rely on statistical testing to
reveal the underlying correlation of the variables with the model’s residuals: Given the
fact that the HT estimator employs variable information that in between the range of the
FEM and REM, Baltagi et al. (2003) for instance suggest to use a pre-testing strategy
that either converts to a FEM, REM or Hausman-Taylor type model depending on the
underlying characteristics of the variable correlation in focus. The estimation strategy
centers around the standard Hausman (1978) test, which has been evolved as a standard
tool to judge among the use of the REM vs. FEM in panel data settings.
However, the Hausman test needs clear underlying assumptions about the consistency
of estimators in comparing either the REM or HT approach with the FEM. Though the
latter serves indeed as a consistent benchmark, Ahn & Low (1996) argue that the test
statistics is only capable in comparing the parameter estimates of time-varying variables
and not time fixed ones. The authors therefore reformulate the Hausman test in a more
general framework and show that the original setup incorporates and tests only a very
limited set of moment conditions among a much broader pool of IV-set candidates. The
latter reformulation of the Hausman tests rests on the Sargan (1958) / Hansen (1982)
statistic of testing for overidentifying restrictions. Together with the closely related CStatistic derived by Eichenbaum et al. (1998), which allows for testing single instrument
validity rather than full IV-sets, the Hansen-Sargen overidentification test may thus be
seen as a more powerful tool to guide IV selection in the HT approach compared to the
standard Hausman test.
As an alternative to IV estimation different ’two-step’-type estimators have been proposed recently: Plümper & Tröger (2007) for instance set up an augmented FEM model
that also allows for the estimation of time-fixed parameters. Their model - labeled Fixed
Effects Vector Decomposition (FEVD) - may be seen as a rival specification for the HT
approach in estimating the full parameter space in the model including both time-varying
and time-fixed regressors. The idea of FEM based two step estimators is thereby to first
run a consistent FEM model to obtain parameter estimates of the time-varying variables.
Using the regression residuals as a proxy for the unobserved individual effects in a second
step this proxy is regressed against the set of time-fixed variables to obtain parameter
values for the latter. Since this second step includes a ’generated regressand’ (Pagan,
5
1984) the degrees of freedom have to be adjusted to avoid an underestimation of standard
errors (see e.g. Atkinson & Cornwell, 2006, for a comparison of different bootstrapping
techniques to correct standard errors in these settings). Though it is typically argued that
one main advantage of these non-IV estimators is their freedom of any arbitrary classification of hand side regressors as being endogenous or exogenous, as we will show latter
on two-step estimators such as the FEVD also rests upon an implicit choice that may
impact upon estimator consistency and efficiency.
Giving the growing number of empirical applications of the latter non-IV approach
(see e.g. Belke & Spies, 2008, Caporale et al., 2008, Etzo, 2007, and Krogstrup & Wälti,
2008, among others),2 a systematic comparison of the HT instrumental variable approach
with the non-IV FEVD is of great empirical interest. However, there are relatively few
existing studies comparing the two-step estimators with the Hausman-Taylor IV approach
in a Monte-Carlo simulation experiment (in particular Plümper & Tröger, 2007, as well
as Alfaro, 2006), which show somewhat heterogeneous results concerning estimator superiority. Moreover, in these studies as well as the broader Monte Carlo based evidence
on the Hausman-Taylor estimator (see e.g. Ahn & Low, 1996, Baltagi et al., 2003), the
empirically unsatisfactory assumption is made that the true underlying correlation between right hand side variables and error term is known. Our approach therefore explicitly
offsets from earlier simulation studies and allows for the existence of imperfect knowledge
in the HT model estimation with IV selection based on different model/moment selection
criteria (see e.g. Andrews, 1999, Andrews & Lu, 2001). The latter combines information
from the Hansen-Sargan overidentification test and time-series information-criteria such
as AIC/BIC. This allows for an empirical comparison of the HT and FEVD (two-step)
estimators’ performance, which comes much closer to the true estimation problem researchers face in applied modelling work in terms of ’To IV or not IV?’.
The remainder of the paper is organized as follows: In section 2 we briefly outline the
general panel data model of interest and describe the two alternative estimation strategies. In section 3 we discuss the Hausman test and the Hansen-Sargan overidentification
test as model/moment selection criteria. Section 4 presents the design and results of our
Monte Carlo simulation experiment. For the Monte Carlo simulations we propose different model/moment selection algorithms for the HT model based on statistical criteria.
Section 5 adds an empirical application to trade estimates in a gravity model context for
German regions (NUTS1-level) within the EU27. Section 6 finally concludes.
2 The FEVD approach is available as a user-written Stata routine upon request from the authors, which additionally
facilitates a widespread empirical usage. Searching for the term ”Fixed Effects Vector Decomposition”by now gives almost
200 entries in Google.
6
2
The model and panel data estimation techniques
We consider a general static (one-way) panel data model of the form
yit = βXit + γZi + uit with: uit = µi + νit
(1)
where i = 1, 2, . . . , N is the cross-section dimension and t = 1, 2, . . . , T the time dimension of the panel data. Xit is a vector of time-varying variables, Zi is a vector of
time invariant right hand side variables, β and γ are coefficient vectors. The error term
uit is composed of two error components, where µi is the unobservable individual effect
and νit is the remainder error term. µi and νit are assumed to be iid(0, σµ ) and iid(0, σν )
respectively.
Standard estimators for the panel data model in eq.(1), which control for the existence
for individual effects, are the FEM and REM approach. To assess the main difference
between the two estimators it is helpful to point out the underlying assumptions about
the correlation of Xi,t and Zi with µi and νi,t respectively:
REM FEM
E(Xi,t µj )
=0
= 0
∀i, j, t (2)
E(Xi,t νj,s )
=0
=0
∀i, j, t, s
E(Zi µj )
=0
= 0
∀i, j
E(Zi νj,s )
=0
=0
∀i, j, s
While both estimators generally assume that the all r.h.s. regressors are exogenous with
respect to the remainder error term νi,t , the assumption about respective variable correlation with the unobservable individual effect µi differs significantly: The REM assumes
strict exogeneity of all regressors, while the FEM approach leaves the correlation as unknown and thus potentially different from zero.
As Boumahdi & Thomas (2008) show, based on the above stated assumptions about
the underlying variable correlation with the error term most panel data estimators for
eq.(1) can be written in terms of a general orthogonality (moment) condition in matrix
form as:
1
1
E[S (tU )] = 0 S tY = S tW δ
(4)
N
N
where S is the (N T xL) instrument matrix, t the (N T xN T ) transformation operator,
7
W = (X, Z) and δ = (β , γ ). If we define Q = IN T − B and B = IN (1/T )eT eT as within
and between matrix operators respectively, where IN T and IN are identity matrices of
order N T and N respectively and eT is a T vector of ones, we can then write the FEM
and REM model in the notation of eq.(2) as:
• FEM: S = X, T = Q and δ̂F EM = (X QX)−1 X QY
• REM: S = W , T = Ω−1/2 and δ̂REM = (W Ω−1 W )−1 W Ω−1 Y
where Ω is the variance-covariance matrix of the error term U . While the FEM uses
only deviations from group means of X as instruments (the vector Z cancels out), the
REM uses all available information in terms both deviations and group means as valid
instrumental variables. Therefore the FEM is always a consistent estimator estimator, if
there is any correlation of one of the variables with the unobservable individual effects
(µi ). On the contrary, if the strict exogeneity assumption of the REM approach is fulfilled,
the latter is more efficient than the FEM since it employs more information for estimation.
Testing for the validity of FEM/REM assumptions is typically done based on the Hausman
(1978) exogeneity test, which we will describe more in detail in section 3.
As eq.(2) shows, choosing among the FEM and REM estimator is linked to an ’all or
nothing’ decision with respect to the assumed correlation of right hand side variables with
the error term. However, the empirical truth may often lie in between these two extremes.
This ideas motivates the specification of the Hausman–Taylor (1981) model as a hybrid
version of the FEM/REM using IV techniques. HT approach therefore simply split the
set of time varying variables into two subsets Xi,t = [X1i,t , X2i,t ] with:
E(X1i,t µj ) = 0, ∀i, j, t
(5)
E(X1i,t νj,s ) = 0, f oralli, j, t, s
E(X2i,t µj ) = 0, ∀i, j, t
E(X2i,t νj,s ) = 0, ∀i, j, t, s
X1 are supposed to be exogenous w.r.t µi and νi,t . X2 variables are correlated with µi
and thus endogenous w.r.t. the unobserved individual effects.3 An analogous classification
is done for the set of time–fixed variables Zi = [Z1i , Z2i ]:
E(Z1i µj ) = 0, ∀i, j
(6)
3 Here we use the terminology of ’endogenous’ and ’exogenous’ to refer to variables that are either correlated with the
unobserved individual effects µi or not. An alternative classification scheme used in the panel data literature classifies
variables as either ’doubly exogenous’ with respect to both error components µi and νi,t or ’singly exogenous’ to only ν.
We use these two definitions interchangeably here.
8
E(Z1i νj,s ) = 0, ∀i, j, s
E(Z2i µj ) = 0, ∀i, j
E(Z2i νj,s ) = 0. ∀i, j, s
Note that the presence of X2 and Z2 is the cause of the bias in the standard REM
approach. The resulting augmented HT model can be written as:
yi,t = α + β1 X1i,t + β2 X2i,t + γ1 Z1i + γ2 Z2i + ui,t .
(7)
The idea of HT model is to find appropriate internal instruments to estimate all model
parameters. Thereby, deviations from group means of X1, X2 serve as instruments for
X1 and X2 (in the logic of the FEM), Z1 serve as their own instruments and group
means of X1 are used to instrument the time-fixed Z2. The FEM and the REM can be
derived as special versions of the HT model, namely when all regressors are correlated
with the individual effects the model reduces to the FEM. For the case that all variables
are exogenous (in the sense of no correlation with the individual effects) the model takes
the REM form.
In empirical terms the HT model is typically estimated by GLS and throughout the paper we use a generalized instrumental variable (GIV) approach, which can be summarized
as:4
• HT: S = [QX1, QX2, P X1, Z1], T = Ω−1/2 and
δ̂HT −GIV = [W Ω−1 S(S Ω−1 S)−1 S Ω−1 W ]−1 W Ω−1 S(S Ω−1 S)−1 S Ω−1 y
(8)
The GIV estimator originally proposed by White (1984) applies 2SLS to the GLS
filtered model (including the instruments) as:5
+ β X2
+ γ Z1
+ γ Z2
+ ũ ,
ỹi,t = α̃ + β1 X1
i,t
i,t
i
i
i,t
2
1
2
(9)
where ỹi,t denotes the following transformation for a variable ỹi,t = Ω̂−1/2 yi,t . Ω̂ is the
estimated variance-covariance matrix of the model defined as Ω̂ = σ̂µ2 (IN ⊗JT )+σ̂ν2 (IN ⊗IT )
with σ̂ν2 = [(û Qû)]/N (T − 1) and σ̂µ = [(û P û) − (N σ̂ν2 )]/N T being consistent estimates
of the variance terms of the error components (for details see e.g. Baltagi, 2008). Finally,
the order condition for the HT estimator to exist is k1 ≥ g2 . That is, the total number of
4 On
has to note that the IV set is based on the interpretation of Breusch et al. (1989).
also has to note that the HT model can also be estimated based on a slightly different transformation, namely the
filtered instrumental variable (FIV) estimator. The latter transforms the estimation equation by GLS but uses unfiltered
instruments. However, both approaches typically yield similar parameter estimates, see Ahn & Schmidt (1999).
5 One
9
time-varying exogenous variables k1 that serve as instruments has to be at least as large
as the number of time invariant endogenous variables (g2 ).6 For the case that (k1 > g2 )
the equation is said to be overidentified and the HT estimator obtained from a 2SLS
regression is generally more efficient than the within estimator (see also Baltagi, 2008).
In empirical application of the HT approach the main points of critique focus on the
arbitrary IV selection in terms of X1/X2 and Z1/Z2 variable classification as well as
the poor small sample properties of IV–methods when instruments are ’weak’ as well
as similar small sample problems of the GLS estimator. Therefore, recent two-step nonIV alternatives such as the Fixed Effects Vector Decomposition (FEVD) by Plümper &
Tröger (2007) have been proposed.7 The goal of the model is to run a consistent FEM
model and still get estimates for the time-invariant variables. The intuition behind the
FEVD specification is as follows: Since the unobservable individual effects capture omitted
variables including the effect of time-invariant variables, it should therefore be possible to
regress a proxy of the individual effects obtained from a first stage FEM regression on the
time-invariant variables to obtain estimates for these variables in a second step. Finally,
the number of degrees of freedom for the use of a ’generated regressand’ in this second
step has to be corrected (e.g. by bootstrapping methods, see Atkinson & Cornwell, 2006).
We can thus sum up the FEVD estimator as:
• FEVD: 1.) Run a standard FEM as described above to get parameter estimates
(β̂F EV D ) of the time-varying variables.
• FEVD: 2.) Use the estimated group residuals as a proxy for the time-fixed individual
effects π̂i obtained from the first step as π̂i = (ȳi − β̂F EM X̄i ) to run a OLS regression
of the explanatory time-invariant variables against this vector to obtain parameter
estimates of the time-fixed variables:
γ̂F EV D = (Z Z)−1 Z π̂
(10)
The residual term from this 2. modelling step η̂i is composed of η̂i = ζi + X̄i (β̂F EM −β),
where ζi = µi + ν̄i and the over-bar indicates the sample period mean for cross-section i
e.g. X̄i = 1/T Tt=1 Xi,t (for details see Atkinson & Cornwell, 2006). Plümper & Tröger
also propose a third (optional) step to control for collinearity between time-varying and
time-fixed right hand side variables in a pooled OLS setup as:
6 The
7 The
total number of IVs in the HT model is 2k1 + k2 + g1 (k1 + k2 from QX1 and QX2, k1 from P X1 and g1 from Z1)
FEVD may be seen as an extension to an earlier model in Hsiao (2003). For details see Plümper & Tröger (2007).
10
yi,t = α + β Xi,t + γ Zi + η̂i + ei,t ,
(11)
In either the 2. or 3. step also standard errors have to be corrected for γ̂F EV D either
asymptotically or by bootstrapping techniques (see Murphy & Topel, 1985, as well as
Atkinson & Cornwell, 2006) to avoid an overestimation of t-values. To sum up, the FEVD
’decomposes’ the vector of unobservable individual effects into a part explained by the
time invariant variables and an error term. Since the FEVD is built on the FEM it yields
unbiased and consistent estimates of the time-varying variables. According to Plümper
& Tröger one major advantage of the FEVD compared to the Hausman-Taylor model is
that the estimator does not require prior knowledge of correlation between the explanatory
variables and the individual effects. Moreover, the specification relies on the robustness of
the FEM and does not need to meet the strong orthogonality assumptions of the REM.
However, estimates of the time invariant variables are only consistent if either the time
invariant variables fully account for the individual effects or the unexplained part of ηi is
uncorrelated with the time-invariant variables. As Caporale et al. (2008) note, otherwise
the FEVD also suffer from omitted variable bias.8 To make this point clear we can write
the FEVD model in terms of the following moment conditions:
E(Xi,t νi,t ) = 0,
(12)
E(Xi,t µi ) = 0,
E(Zi µi ) = 0.
The latter orthogonality condition can be obtained from the definition of ηi above.
Thus, though we are not directly confronted with the choice of classifying variables as
endogenous or exogenous, the estimator itself does rely on an implicit choice: In specifying
the time-varying variables the model follows the generality of the FEM approach, which
assumes that these variables are possibly correlated with the unobservable individual
effects (for estimation purposes deviations from group means are taken which wipe out
the individual effects so that no explicit assumption about the underlying correlation
needs to be stated). With respect to the time invariant variables the estimator assumes
in its simple form that no time-fixed variable (Z) is correlated with the the second step
error term, which is composed of the unobservable individual effects. However, if this
8 A modification of the standard FEVD approach also allows for the possibility to estimate the second step as IV regression
and thus account for endogeneity among
time invariant variables and ηi . Following Atkinson & Cornwell, 2006, we can define
a standard IV estimator as: γ̂F EV D = S Z)−1 S π̂ , where S is the instrument set that satisfies the orthogonality condition
E(Sη) = 0. However, this brings back the classification problem of the HT approach, which we aim to avoid here.
11
implicit (and fixed) choice does not reflect the true correlation between the variables and
the individual effects the estimator may in fact have lower power than the HT approach.
We carefully examine the relative performance of the two estimators in the Monte
Carlo simulations. Reducing the empirical problem to the question of finding a proper
instrument set (leaving the different small sample properties of IV and non-IV methods for
the moment apart), the above results also advice to test IV validity before choosing among
one of the rival estimators based on statistical testing. We will thus turn to statistical
moment selection criteria in the next section.
3
Moment Selection Criteria
To judge among the estimators’ orthogonality assumption different specification tests have
been proposed. The most prominent example is the Hausman (1978) exogeneity test based
on the m-statistic defined as:
m = q̂ (Q̂ − V̂ )−1 q̂,
(13)
with q̂ = δ̂F EM − δ̂REM , Q̂ and V̂ as consistent estimates of the asymptotic covariance
matrices of δ̂F EM and δ̂REM m-statistic has a χ2 -distribution with degrees of freedom equal
to the number of parameter estimates. The test idea rests on the basic assumption that the
REM is generally more efficient than the FEM. However, if the difference between the two
estimators is large, the exogeneity assumptions in the REM are not met and the estimator
is supposed to be misspecified. Thus, under the null hypothesis of the Hausman test both
estimators are assumed to be consistent, but δ̂REM is more efficient than δ̂F EM . Under
the alternative hypothesis only δ̂F EM is consistent, δ̂REM is misspecified. In empirical
application to panel data estimation, e.g. Baltagi et al. (2003) use the Hausman test to
construct a pretest estimator of the following form: In the first case the pretest estimator
reverts to REM if Hausman test for REM vs. FEM is not rejected. If the strong set of
moment conditions of the REM is rejected, next the HT model validity is tested through
a second Hausman test based on HT vs. FEM and if the HT specification is also rejected,
the pretest estimator takes the FEM form.
Ahn & Low (1996) argue that the Hausman test according to eq. (13) has limited power
because it only tests for the consistency and efficiency of β and not γ. Thus, the m-statistic
can only detect misspecification if β̂REM (as testable part of δ̂REM ) becomes substantially
biased and different from β̂F EM , however as Ahn & Low (1996) argue moderate levels of
correlation between Zi and µi are unlikely to cause a significant bias in β̂REM . The authors
therefore conclude that the Hausman test outcomes should be interpreted with caution
12
and propose a more general test setup based on the Sargan (1958) / Hansen (1982) test
for overidentification of moment conditions. In its general GMM form the J-Statistic is
defined as:
J(δ̂EGM M ) = ûS (S Ω̂S)−1 S û ∼ χ2 (k − g)
(14)
where û are 1.step (2SLS) residuals, (k−g) is the number of overidentifying restrictions.
The J-Statistic is the value of the GMM objective function, evaluated at the efficient GMM
estimator δ̂EGM M . In an overidentified model it allows to test whether the model satisfies
the full set of moment conditions. A rejection implies that IVs do not satisfy orthogonality
conditions required for their employment. In its general form the J-Statistic allows for
numerous testing setups depending on the choice of IV set and transformation operator.
Thus, any test upon the J-Statistic can be regarded as a generalisation of the standard
Hausman test. In fact, Ahn & Low (1996) prove that the latter tests the orthogonality
conditions as J(δ̂GLS ) with S = [QX, P X].
If the ’No Conditional Heteroscedasticity’ NCH -condition holds, for any given IV estimator the J-Statistic coincides with the familiar Sargan (1958) statistic as:
J(.) = Sargan =
û P û
û S(S S)−1 S û
∼ χ2 (k − g)
=
σ̂ 2
û û/n
(15)
A nice fact about the Sargan (1958) Statistic is that it has a very intuitive interpretation: That is, since it has an nRu2 form, where Ru2 is the uncentered R-squared and n is
the total number of observations, it can be easily calculated by regressing the residuals of
the IV regression on the full instrument set S. Since the model fit increases with a higher
correlation of the residuals and the instrument set, this signals doubts for the validity of
the model’s underlying orthogonality assumptions.
Based on the J-Statistic we can also derive a test for a subset of valid moment conditions
for instrument choice rather than the full IV set S. This so-called C-Statistic has been
proposed by Eichenbaum et al. (1988) and tests the following hypothesis:
H0 : E(S1i ui ) = 0
and
(S2i ui ) = 0
(16)
H0 : E(S1i ui ) = 0
and
(S2i ui ) = 0
(17)
where S is divided into S1 and S2 . The latter subset contains those instruments, for
which exogeneity shall be tested. Under the null hypothesis both sub sets are orthogonal
to the error term, under the alternative hypothesis only S1 is exogenous. Numerically, the
C-Statistic can be derived as the difference of two Hansen-Sargan overidentification tests
with C = J − J1 ∼ χ2 (M − M1 ), where M1 is the number of instruments in S1 and M
13
is the total number of IVs. We will make use of the above defined statistical criteria to
guide moment condition selection in the HT case assuming imperfect knowledge about the
correlation of r.h.s variables with the residuals for the Monte Carlo simulation experiment.
4
Monte Carlo Simulations
We specify a Monte Carlo simulation experiment in the spirit of Im et al. (1999) and
Baltagi et al. (2003). We use a static one-way model as in eq.(1) including 4 time-varying
(X) and 3 time-fixed (Z) regressors of the form:
yi,t = β11 x11 ,i,t + β12 x12 ,i,t
(18)
+β21 x21 ,i,t + β22 x22 ,i,t
+γ11 z11 ,i + γ12 z12 ,i
+γ21 z21 ,i + ui,t ,
with: ui,t = µi + νi,t
where x11 and x12 are assumed to be uncorrelated with the error term, while x21 and
x22 are correlated with µi . Analogously, z21 is correlated with the error term. The latter
is composed of the unobserved individual effects (µi ) and remainder disturbance (νi,t ).
The time-varying regressors x11 , x12 , x21 , x22 are generated by the following autoregressive
process:
xnm ,i,t=1 = 0 with n, m = 1, 2
(19)
x11 ,i,t = ρ1 x1i,t−1 + δi + ξi,t for t = 2, . . . , T
(20)
x12 ,i,t = ρ2 x2i,t−1 + ψi + ωi,t for t = 2, . . . , T
(21)
x21 ,i,t = ρ3 x3i,t−1 + µi + τi,t for t = 2, . . . , T
(22)
x22 ,i,t = ρ4 x4i,t−1 + µi + λi,t for t = 2, . . . , T
(23)
For the time-fixed regressors z11 , z12 , z21 we analogously define:
z11 ,i = 1
(24)
z12 ,i = g1 ψi + g2 δi + κi
(25)
z21 ,i = µi + δi + ψi + i
(26)
The variable z11 ,i simplifies to a constant term, z21 ,i is the endogenous time-fixed regressor since it contains µi as r.h.s. variable, the weights g1 and g1 in the specification of
14
z12 ,i control for the degree of correlation with the time-varying variables x11 ,i,t and x12 ,i,t .9
The remainder innovations in the data generating process are defined as follows:
νi,t ∼ N (0, σν2 )
(27)
µi ∼ N (0, σµ2 )
(28)
δi ∼ U (−2, 2)
(29)
ξi,t ∼ U (−2, 2)
(30)
ψi ∼ U (−2, 2)
(31)
ωi,t ∼ U (−2, 2)
(32)
τi,t ∼ U (−2, 2)
(33)
λi,t ∼ U (−2, 2)
(34)
i ∼ U (−2, 2)
(35)
κi ∼ U (−2, 2)
(36)
Except µi and νi,t , which are drawn from a normal distribution with zero mean and
variance σµ2 and σν2 respectively, all innovations are uniform on [-2,2]. For µi , δi , ψi , i , κi
the first observation is fixed over T . With respect to the main parameter settings in the
Monte Carlo simulation experiment we set:
• β11 = β12 = β21 = β22 = 1
• γ12 = γ21 = 1
• ρ1 = ρ2 = ρ3 = ρ4 = 0.7
All variable coefficients are normalized to one, the specification of ρ < 1 assures that
the time-varying variables are stationary. We also normalize σν equal to one and define
a load factor ξ determining the ratio of the variance terms of th e error components as
ξ = σµ /σν . ξ takes values of 2;1 and 0.5. We run simulations with different combinations in
the time and cross-section dimension of the panel as N = (100, 500, 1000) and T = (5, 10).
All Monte Carlo simulations are conducted with 500 replications for each permutation in y
and u. As in Arellano & Bond (1991) we set T = T + 10 and cut off first 10 cross-sections,
which gives a total sample size of N T observations.
9 We
vary g1 and g2 on the interval [-2,2]. The default is g1 = g2 = 2.
15
We apply the FEVD and Hausman-Taylor estimators.10 As outlined above, one drawback in earlier Monte Carlo based comparisons between the HT model and rival non-IV
candidates was the strong assumption made for IV selection in the HT case, namely
that true correlation between r.h.s. variables and the error term is known. However, this
may not reflect the identification and estimation problem in applied econometric work
and Alfaro (2006) identifies it as one of the open questions for future investigation in
Monte Carlo simulations. We therefore account for the HT variable classification problem
by implementing algorithms from ’model selection criteria’-literature, which combine information from Hansen-Sargan overidentification test for moment condition selection as
outlined above and time-series information-criteria. Following Andrews (1999) we define
a general model selection criteria (MSC) based on IV estimation as
M SCn (m) = J(m) − h(c)kn
(37)
where n is the sample size, c as number of moment conditions selected by model m
based on the Hansen-Sargan J-Statistic J(m), h(.) is a general function, kn is a constant
term. As eq.(37) shows, the model selection criteria centers around the J-Statistic outlined
in section 3. The second part in eq.(37) defines a ’bonus’ term rewarding models with
more moment conditions, where the form of function h(.) and the constants (kn :≥ 1)
are specified by the researcher. For empirical application Andrews (1999) proposes three
operationalizations in analogy to model selection criteria from time series analysis:
• MSC-BIC: J(m) − (k − g)ln n
• MSC-AIC: J(m) − 2(k − g)
• MSC-HQIC: J(m) − Q(k − g)ln ln n with Q = 2.01
where (k − g) is the number of overidentifying restrictions, and depending on the form
of the ’bonus’ term, the MSC may take the BIC (Bayesian), AIC (Akaike) and HQIC
(Hannan Quinn) form. 11 We apply all three information criteria in the Monte Carlo
simulations motivated by the results in Andrews & Lu (2001) and Hong et al. (2003) that
the superiority of one of the criteria over the others in terms of finding consistent moment
conditions may vary with the sample size.12 For each of these MSC criteria we specify the
following algorithms:
10 For the FEVD estimator we employ the Stata routine xtfevd written by Plümper & Tröger (2007), the HT model is
implemented using the user written Stata routine ivreg2 by Baum et al. (2003).
11 The BIC criterion was introduced by Schwartz (1978), the AIC by Akaike (1977) and the HQIC by Hannan & Quinn
(1979).
12 Generally, the MSC-BIC criterion is found to have the best empirical performance in large samples, while the MSC-AIC
outranks the other criteria in small sample settings, but performs poor otherwise.
16
1. Unrestricted form: For all possible IV combinations out of the full IV-set S=(QX1,
QX2, PX1, PX2, Z1, Z2), which satisfy the order condition k1 > g2 (giving a total
number of 42 combinations), we calculate the value of the MSC criterion (for the
BIC, AIC and HQIC separately) and choose that model as final HT specification,
which has minimum MSC value over all candidates.
2. Restricted form: This algorithm follows the basic logic from above, but additionally
puts the further restriction that only those models serves as MSC candidates for
which the p-value of the J-Statistic is a above a critical value Ccrit. , which we set to
Ccrit. = 0.05 to be sure that the selected moment conditions are true in terms of statistical pre-testing. The restricted version thus follows the advice of Andrews (1999)
to ensure that the parameter space incorporates only information, which assumes
that certain moment conditions are correct.
We present flow charts of the restricted and unrestricted MSC based search algorithm
in figure 1.
<<< insert Figure 1 about here >>>
As Andrews (1999) argues, the above specified model selection criteria is closely related
to the C-Statistic approach by Eichenbaum et al. (1988) to test whether a given subset of
moment conditions is correct or not. Thus alternatively to the above described algorithms,
we specify a downward testing approach based on the C-Statistic: Here we start from
the HT model with with full IV set in terms of the REM moment conditions as S1 =
(QX1, QX2, P X1, P X2, Z1, Z2). We calculate the value of the J-Statistic for the model
with IV-set S1 and compare its p-value with a predefined critical value Ccrit. , which we
set in line with the above algorithm as Ccrit. = 0.05. If PS1 > Ccrit. we take this model as
a valid representation in terms of the underlying moment conditions. If not, we calculate
the value of the C-Statistic for each single instrument in S1 and exclude that instrument
from the IV-set that has the maximum value of the C-statistic.
We then re-estimate the model based on the IV-subset S2 net of the selected instrument
with the highest C-Statistic and again calculate the J-Statistic and its respective p-value.
If PS2 > Ccrit. is true, we take the HT-model with S2 as final specification and otherwise
again calculate the C-Statistic for each instrument to exclude that one with the highest
value. We run this downward testing algorithm for moment conditions until we find a
model that satisfies PS. > Ccrit. or at the most until we reach the IV-sets Sn to Sm ,
where the number of overidentifying restrictions (k − g) = 1, since the J-Statistic is not
defined for just identified models. Out of Sn to Sm we then pick the model with the
17
lowest J-Statistic value. The C-Statistic based model selection algorithms is graphically
summarized in figure 2.
<<< insert Figure 2 about here >>>
Turning to the results of the Monte Carlo simulations, detailed information about the
estimators’ performance for the above specified parameter settings in N , T and ξ are
reported in table A.1 to table A.6.13 Since we are interested in consistency and efficiency
of the respective estimators, we compute the empirical bias, its standard deviation and
the root mean square error (rmse). The bias is defined as
bias(δ̂) =
M
(δ̂ − δtrue )/M,
(38)
m=1
where m = (1, 2, . . . , M ) is the number of simulation runs. Next to the standard
deviation of the estimated bias we also calculate the root mean square error, which puts
a special weight on outliers, as:
2
M
rmse(δ̂) = (δ̂ − δtrue )/M .
(39)
m=1
We first take a closer look at the individual parameter estimates for the parameter
settings N = 1000, T = 5 and ξ = 1, which are typically assumed in the standard Panel
data literature building on the large N , small T data assumption. In figure 3 we plot
Kernel density distributions for all regression coefficients for the following three estimators:
i.) the FEVD, ii.) the HT model with perfect knowledge about the underlying variable
correlation with the error term and iii.) the HT model based on the MSC-BIC algorithm
(in its restricted form). The latter shows on average the best performance among all
HT estimators with imperfect knowledge about the underlying data correlation - closely
followed by the C-Statistic based model selection algorithm.
For the coefficients of the two exogenous time-varying variables β11 and β12 all three
estimators give almost unbiased results centering around the true parameter value of one.
The standard deviation and rmse are the smallest for the HT model with perfect knowledge
about the underlying data correlation, followed by the algorithm based HT estimators. The
FEVD has a slightly higher standard deviation and rmse. For the estimated coefficients
of the endogenous time-varying variables β21 and β22 the HT and FEVD give virtually
13 In the following we do not present simulation results for the constant term z
11 in the model. Results can be obtained
from the author upon request.
18
identical results, while the HT based MSC-BIC in figure 1 is slightly biased for β21 but
comes closer to the true parameter value for the parameter β21 .
To sum up, though there are some minor differences among the three reported estimators for the time-varying variables in figure 1, the overall empirical discrepancy is rather
marginal. This picture however radically changes for the Monte Carlo simulation results of
the time-fixed variable coefficients γ12 and γ21 : Here only the HT model with the ex-ante
correctly specified variable correlation gives unbiased results for both the exogenous (γ12 )
and endogenous variable (γ21 ). Both the FEVD and HT model based on the MSC-BIC
have difficulties in calculating these variable coefficients correctly, while the bias of the
FEVD is somewhat lower than for the MSC-BIC Hausman-Taylor model in both cases.
Especially for γ21 exclusively all HT based model selection algorithms have a large bias/standard deviation as well as a high rsme relative to the HT with perfect knowledge
abot the variable correlation with the error term. The FEVD has a significant bias (approximately 50 percent higher than the standard HT) but compared to the MSC-BIC
based specification a lower bias/standard deviation.
<<< insert Figure 3 about here >>>
Turning to the small sample properties for the above mentioned estimators we additionally plot Kernel density plots for the parameter settings N = 100, T = 5, ξ = 2. Here
the results in figure 4 show that the MSC-BIC based HT model is already more biased
compared to the standard HT and FEVD for the parameter estimates of the time-varying
variables β11 , β12 , β21 and β22 , where in all cases the bias is the smallest for the FEVD.
With respect to the rmse the smallest value for β11 and β12 is given by the C-Statistic
based HT model, while FEVD and the standard HT model perform best for β21 and β22 .
For the time-fixed variables again the FEVD and the MSC-BIC based HT model have a
significant bias, while the HT model with perfect knowledge about the underlying variable
correlation comes on average much closer to the true parameter value (in particular for
γ12 ). However, as already observed in Plümper & Tröger (2007) the standard deviation
of the latter estimator is much higher compared to the other two estimators. This leads
to the result that in terms of the rmse the FEVD performs better than the standard
HT in these settings (for both γ12 and γ21 ), although it shows a larger bias compared to
the latter. The results in figure 4 indicate that the HT instrumental variable approach is
rather inefficient in small sample settings, though the average bias is small.
<<< insert Figure 4 about here >>>
19
A specific problem of the MSC-BIC based HT model in small sample settings is shown
in figure 5. The Kernel density plot for the coefficient γ21 of the endogenous time-fixed
variables reveals a ’duality’ problem for the search algorithm based estimator, which significantly increases with small values for ξ. Different from the standard HT and FEVD
estimators the MSC-BIC based HT model shows a clear double peak for parameter estimates of γ̂21 , with one peakaround the true coefficient value of one and a second significantly
biased one. This kind of duality problem with a possibly poor MSC based estimator performance has already been addressed in Andrews (1999) for those cases where there are
typically two or more selection vectors that yield MSC values close to the minimum and
parameter estimates that differ noticeably from each other. As the histogram in figure 6
shows, this is indeed the problem for the MSC-BIC based HT model: Based on the Monte
Carlo simulation runs with 500 reps. the algorithm tends to pick two dominant IV-sets
from which one has the (inconsistent) REM form with a full instrument list, while only the
second one consistently excludes Z21 from the instrument list. This results may be seen
as a first indication that in small samples and a small proportion of the total variance of
the error term due to the random individual effects (through low values of ξ), J-statistic
based IV selection may have a low power and yield inconsistent results.
<<< insert Figure 5 and Figure 6 about here >>>
Turning from a comparison of single variable coefficients to an analysis of overall measures of model bias and efficiency for an aggregated parameter space, we compute NOMAD and NORMSQD values, where the NOMAD (normalized mean absolute deviation)
computes the absolute deviation of each parameter estimate from the true parameter, normalizing it by the true parameter and averaging it over all parameters and replications
considered. The NORMSQD computes the mean square error (mse) for each parameter,
normalizing it by the square of the true parameter, averaging it over all parameters and
taking its square root (for details see Baltagi & Chang, 2000). Both overall measures
are thus extensions to the single parameter bias and rmse statistics defined above. We
compared the FEVD model with the standard HT model and the algorithm based HT models using the C-Statistic approach, as well as the MSC-BIC1, MSC-HGIC1, MSC-AIC1
(where the index 1 denotes that all are based on the restricted specification).
We start to report surface plots for the two parameter aggregates 1.) time-varying and
2.) time-fixed variable coefficients over all different settings in our Monte Carlo simulation
experiment. The choice of aggregation is motivated by the above findings that the results
significantly differ with respect to time-varying and time-fixed variable coefficients. Figure
7 and figure 8 plot NOMAD and NORMSQD values for the time-varying coefficients β11
20
to β22 . The figures show that in terms of bias the algorithm based HT models show a
significant small sample problem, while the bias of the FEVD and standard HT is rather
small for different combinations in the time and cross-section dimension of the data. In
terms of the rmse all estimators (both IV and non-IV) show a significant decrease in the
rmse value with increasing number of total observations N T .
With respect to the rmse the standard HT model performs best, closely followed by
the FEVD. As Plümper and Tröger (2007) already expected without explicit testing, the
efficiency of the HT approach significantly reduces if the underlying assumption about
the variable correlation with the error term is not known. Only the MSC-BIC comes
close to the FEVD as second best estimator. Another interesting finding is that figure
7 and 8 both show distinct spikes alongside the overall trend of increasing estimator
power with larger sample size. These spikes (in particular for the search algorithm based
HT models) are induced by low values for ξ = 0.5, which significantly deteriorate the
estimators performance in terms of bias and rmse. This result comes close to findings in
Baltagi et al. (2003), where for small values of ξ their pretest estimator reverts to the
REM specification although the underlying data structure implies a correlation of r.h.s.
variables with the individual effects in the sense of a HT world. Baltagi et al. (2003) report
misleading inference of the pretest estimator in this case.
Turning to the NORMAD and NORMSQD values for the time-fixed regressors, figure 9
and 10 show that average bias and rmse are roughly constant over different combinations in
the time and cross-section dimension of the data, where the smallest bias is obtained from
the standard HT model with perfect knowledge about the underlying variable correlation.
On average the bias of the FEVD is significantly higher than for the standard HT. Also the
performance of the algorithm based HT models is rather poor. Contrary to the estimation
bias, in terms of rmse the FEVD clearly performs best. The high rmse for HT models
with imperfect knowledge about the underlying variable correlation especially in small
sample settings, can be explained by the above identified duality problem of this statistical
approach. The standard HT model also has small sample problems, but comes close to
the rmse value of the FEVD estimator for a larger sample size.
<<< insert Figure 7 to Figure 10 about here >>>
To sum up, we finally average the NOMAD and NORMSQD values over all variable
coefficients (see table 1) and plot the results in figure 11. The figure shows that the HT
model with perfect knowledge about the underlying variable correlation has the lowest
NOMAD value, with the FEVD having two times and algorithm based HT specification
even three times higher values for the average bias over all model coefficients. For the
21
latter the C-Statistic based model selection criteria performs slightly better than the
MSC based estimators. Contrary, with respect to the NORMSQD by far the best model
is the non-IV FEVD. The difference between the standard HT model and the algorithm
based specification is rather low. This broad picture indicates that the HT instrumental
variable model is a consistent estimator given perfect knowledge about the true underlying
correlation between the r.h.s. variables and the error term. However, when one has to rely
on statistical criteria to guide moment condition selection the empirical performance for
the specific setup in the Monte Carlo simulation design is considerably lower. This in turn
speaks in favor of using non-IV two step estimators such as the FEVD, which although it
may introduce a moderate bias in estimating the time-fixed variables, has the lowest rmse
due to its robust OLS estimation approach compared to the IV based rival HT estimators.
<<< insert Table 1 and Figure 11 about here >>>
5
Empirical application: Trade estimates for German regional
data
Given the above findings from our Monte-Carlo simulation experiment in this section we
aim to consider the empirical performance of the FEVD and HT model in an empirical
application estimating gravity type models. We take up the research question in Alecke et
al. (2003 & 2007) and specify trade equations for regional entities. In particular, we aim
to re-estimate gravity models for export flows among German states (NUTS1-level) and
its EU27 trading partners using an updated database for the period 1993-2005. We are
particularly interested in quantifying the effects of time-fixed variables including geographical distance as a general proxy for trading costs as well as a set time-fixed 0/1-dummies
for border regions, the East German states as well as the specific trade pattern with the
CEEC countries.14 Earlier evidence in Alecke et al. (2007) has shown that there is a considerable degree of heterogeneity for these time-fixed variables among different estimators
(similar evidence is given in Belke & Spies, 2008). The empirical export model we focus
on has the following form:15
14 The CEEC aggregate includes Hungary, Poland, the Czech Republic, Slovakia, Slovenia, Estonia, Latvia, Lithuania,
Romania and Bulgaria.
15 Results for an import equation with qualitatively similar results can be obtained from the author upon request.
22
log(EXijt ) = α0 + α1 log(GDPit ) + α2 log(P OPit ) + α3 log(GP Djt ) + α4 log(P OPjt )
+α5 log(P RODit ) + α6 log(DISTij ) + α7 SIM + α8 RLF
(40)
+α9 EM U + α10 EAST + α11 BORDER + α12 CEEC,
where the index indicates German regional exports from region i to country j for time
period t and imports to German state i from country j respectively. The variables in the
model are defined as follows16 :
– EX = Export flows from region i to country j
– GDP = Gross domestic product in i and j respectively
– P OP = Population in i and j
– P ROD = Labour productivity in i and j
– DIST = Geographical distance between state/national capitals
– SIM = Similarity index defined as: log(1 −
2
GDPi,t
GDPi,t +GDPj,t
−
2
GDPj,t
)
GDPi,t +GDPj,t
GDPi,t
– RLF = Relative factor endowments in i and j defined as: log P OPi,t
−
GDPj,t
P OPj,t
– EM U = EMU membership dummy for i and j
– EAST = East German state dummy for i
– BORDER = Border region dummy between i and j
– CEEC = CEE country dummy for j
The estimation results are shown in table 2. We particularly focus in the FEVD and
HT estimates for the variable log(DISTij ) as well as the time-fixed dummies EAST ,
BORDER and CEEC. The HT approach rests on the C-Statistic based downward testing
approach to find a consistent set of moment conditions.
Turning to the results, in line with our Monte Carlo simulation results both the FEVD
and HT estimators are very close in quantifying the time-varying variables in the gravity
model for German regional export activity. As expected from its theoretical foundation
(see e.g. Egger, 2000, Feenstra, 2004) both home and foreign country GDP have a positive and significant influence on German export activity, indicating that trade increases
with absolute higher income levels. Moreover, also home region productivity (defined as
GDP per total employment) is found to be statistically significant and highly positive,
16 Further
details can be found in the data appendix in table A.7.
23
|
which in turn can be interpreted in line with recent findings based on firm-level data (see
e.g. Helpman et al., 2003, or Arnold & Hussinger, 2006, for the German case) that the
degree of internationalization of home firms (both trade and FDI) increases with higher
productivity levels. The interpretation of the population variable in the gravity model
is less clear cut: Both the FEVD and HT estimator find a positive coefficient sign for
foreign population, which can be interpreted in favour of the market potential approach
indicating that German export flows are higher for population intense economies. Also for
the GDP based interaction variable SIM (definition see above) the two estimators show
similar results.
However, as already observed throughout the Monte Carlo simulation experiment for
the time-fixed variables the estimators show a considerable degree of heterogeneity. In
our export model the C-Statistic based HT approach finds a coefficient for the distance
variable (-1,73) that is almost twice as large as the respective coefficient in the in POLS,
REM and FEVD (-0.97) case. A similar difference between FEVD and HT model results
were also found in Belke & Spies (2008) for EU wide data (the authors report coefficients
for the distance variable in the HT case as -1,83 compared to -1,39 in the FEVD case).
Without the additional knowledge from the above Monte Carlo simulation experiment,
we could hardly answer the question whether this discrepancy among estimators either
indicates an upward bias of the HT model given the fact that (for national data) the
parameter estimate for the distance variable typically ranges between -0.9 to -1.3 (see e.g.
Disdier & Head, 2008 as well as Linders, 2005) or whether the use of smaller regional
entities serves as a better proxy for geographical distance thus gives a more accurate
estimate for trade costs (which may be possibly higher).
However, in the light of the Monte Carlo simulation results together the typical range
of national estimates it seems plausible to rely on the FEVD estimation results, although
the HT model passes the Hansen/Sargan overidentification test (treating geographical
distance as endogenous). Also for the further time-fixed dummy variables in the model the
FEVD estimates show more reliable coefficient signs than the HT model: That is, we would
expect the border dummy to be positive as e.g. found in Lafourcade & Paluzie (2005) for
European border regions. Also, German-CEEC trade was persistently found to be above
its ’potential’ in a couple of earlier studies such as Schumacher & Trübswetter, 2000, Buch
& Piazolo, 2000, Jakab et al., 2001), Caetano et al., 2002 as well as Caetano & Galego,
2003). In both cases the FEVD estimates are thus more in line with recent empirical
findings than the HT IV-estimation. For the FEVD results we may thus finally conclude
that the obtained coefficients for time-fixed variables better reflect similar evidence in
recent empirical work.
24
<<< insert Table 2 about here >>>
6
Conclusion
In this paper we have performed a Monte Carlo simulation experiment to compare the
empirical performance of IV and non-IV estimators for a regression setup, which includes
time-fixed variables as right hand side regressors and where endogeneity matters. We
define the latter as any correlation between the r.h.s. variables with the model’s error term.
In specifying empirical estimators we focus on the Hausman-Taylor (1981) Instrumental
Variable (IV) model both with perfect and imperfect knowledge about the underlying
variable correlation with the model’s residuals and non-IV two-step estimators such as
the Fixed Effects Vector Decomposition (FEVD) model recently proposed by Plümper
& Tröger (2007). Our results confirm earlier empirical findings (see e.g. Alfaro, 2006)
that the HT (with perfect knowledge) works better for time-varying, while the FEVD
for time-fixed variables. Averaging over all parameter we find the the HT model (with
perfect knowledge) generally has the smallest bias, while the FEVD show to have a by far
lower root mean square error (rmse) as a general efficiency measure. Especially in small
sample settings our Monte Carlo simulations show that the IV-based HT model has a
large standard deviation and consequently high rmse values.
Additionally, relaxing the assumption of perfect knowledge for the HT model, the
empirical performance of the latter significantly worsens. We compute different algorithms
to select consistent IV-sets centering around the Hansen/Sargan overidentification test
(J-Statistic), however all estimates based on these algorithms generally show a weaker
empirical performance than the non-IV alternative (FEVD). One major drawback of the
HT models with imperfect knowledge is a ’duality’ problem in small sample settings, where
the estimator has difficulties to discriminate between consistent and inconsistent moment
condition vectors. We may thus conclude that IV selection solely based on statistical
criteria has to be treated with some caution.
Future research in this area should especially focus on possible improvements of these JStatistic based algorithms by combining a measure for model consistency with criteria for
IV-relevance and -uniqueness. Hall & Peixe (2000) as well as Jana (2005) e.g. propose to
combine the J-Statistic with the Canonical correlation information criteria. For practical
application Andrews & Lu (2001) additionally recommend to combine statistically driven
model selection with any other information available about the correct parameter and
moment vector. This would speak in favor of using statistical criteria such as the JStatistic in terms of a fine tuning supporting the basic IV selection on grounds of ’economic
25
intuition’ as proposed by Hausman & Taylor (1981). However, the empirical performance
of such a multi-level selection strategy is hardly to test in Monte Carlo simulations and has
to prove its validity in practical application. An alternative choice for applied researcher
are non-IV two-step estimators such as the FEVD by Plümper & Tröger (2007), which
show an on average good performance in our Monte Carlo simulations and yield very
plausible results for the empirical estimation of German regional trade flows using gravity
type models.
26
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30
Figure 1: MSC based model selection algorithm for HT-approach
1: RESTRICTED FORM
2: UNRESTRICTED FORM
HT-Estimation with
Instrument set Sn
(with n= 1,..., N)
HT-Estimation with
Instrument set Sn
(with n= 1,..., N)
J-Statistic
(p-value)
J-Statistic
(p-value)
YES
P > Ccrit. ?
Compute
MSC(Sn)
Compute
MSC(Sn)
NO
MSC(Sn) = {∅}
Select final IV-Set for HT-model as
Select final IV-Set for HT-model as
min(MSC(S 1), …, MSC(S N))
min(MSC(S 1), …, MSC(S N))
31
32
HT-Estimation with
this IV-set S1
YES
P > Ccrit.?
J-Statistic
(p-value)
Start with full IV-set
S1=(QX,PX,Z)
NO
Calculate C-Statistic
for each IV in S1
C-Stat. max
Exclude IV with
HT-Estimation with
IV-subset S2
YES
P > Ccrit.?
J-Statistic
(p-value)
Estimate with
IV-subset S2
NO
BOX 1
Figure 2: C-Statistic based model selection algorithm for HT-approach
Use IV set with
minimum J-Statistic
for (k - g) = 1
J-Statistic for
each Sn to Sm
Repeat steps in BOX 1
until an IV set is found
that satisfies Ccrit. or
at the most until we
reach IV sets Sn to Sm
with (k - g) = 1
25
20
15
.95
FEVD
1
HT
Parameter beta11
1.05
1.05
.95
FEVD
Monte Carlo Simulation with 500 reps.
.9
1
HT
Parameter beta22
Monte Carlo Simulation with 500 reps.
HT−BIC
1.1
HT−BIC
.95
FEVD
FEVD
.6
Monte Carlo Simulation with 500 reps.
.4
1
HT
.8
HT
1
1.05
Parameter gamma12
Monte Carlo Simulation with 500 reps.
.9
Parameter beta12
1.1
HT−BIC
1.2
HT−BIC
.95
FEVD
1
FEVD
1.5
Monte Carlo Simulation with 500 reps.
.5
1
HT
1.05
HT
2
Parameter gamma21
Monte Carlo Simulation with 500 reps.
.9
Parameter beta21
Figure 3: Kernel density plots for Monte Carlo simulation results with N = 1000, T = 5, ξ = 1
5
0
5
10
15
20
25
0
10
25
20
15
10
5
0
10
5
0
20
15
10
5
0
6
4
2
0
33
1.1
HT−BIC
2.5
HT−BIC
15
.8
.9
1
HT
Parameter beta11
.9
FEVD
1
1.1
Parameter beta22
Monte Carlo Simulation with 500 reps.
FEVD
1.1
1.2
1.2
1.3
HT−BIC
.9
FEVD
0
1
HT
1.1
.5
1
HT
1.5
Parameter gamma12
Monte Carlo Simulation with 500 reps.
.8
Parameter beta12
1.2
2
HT−BIC
Monte Carlo Simulation with 500 reps.
HT
HT−BIC
Monte Carlo Simulation with 500 reps.
FEVD
HT−BIC
FEVD
1
FEVD
0
Monte Carlo Simulation with 500 reps.
−2
HT
1.1
HT
2
Parameter gamma21
Monte Carlo Simulation with 500 reps.
.9
Parameter beta21
Figure 4: Kernel density plots for Monte Carlo simulation results with N = 100, T = 5, ξ = 2
5
2
0
4
6
8
10
0
10
15
10
5
0
5
4
3
1
0
2
8
6
4
2
0
2.5
2
1.5
1
.5
0
34
1.2
HT−BIC
4
HT−BIC
Figure 5: Kernel density plots for Monte Carlo simulation results of γ21 with
N = 100, T = 5, ξ = 1
0
.5
1
1.5
2
Parameter gamma21
−2
0
2
FEVD
HT
Monte Carlo Simulation with 500 reps.
35
4
HT−BIC
Figure 6: Histogram of selected IV-sets for Monte Carlo simulation results of γ21 with
N = 100, T = 5, ξ = 1
HT−BIC
200
250
265
50
Frequency
100
150
185
3
8
13
1
1
1
2
1
10
3
3
0
4
0
5
10
# of IV−Set
Monte Carlo Simulation with 500 reps.
36
15
20
(N=1000; T=5, xi=1)
(N=1000; T=10, xi=2)
(N=1000; T=10, xi=1)
(N=1000; T=10, xi=0.5)
(N=1000; T=5, xi=2)
VD
FE
T
1
IC
A
1
TIC
H
Q
H
1
TH
IC
B
Tat
st
C
TH
H
H
37
(N=1000; T=5, xi=0.5)
(N=500; T=10, xi=2)
(N=500; T=10, xi=1)
(N=500; 10=5, xi=0.5)
(N=500; T=5, xi=2)
(N=500; T=5, xi=1)
(N=500; T=5, xi=0.5)
(N=100; T=10, xi=2)
(N=100; T=10, xi=1)
(N=100; T=10, xi=0.5)
(N=100; T=5, xi=2)
(N=100; T=5, xi=1)
(N=100; T=5, xi=0.5)
Figure 7: NOMAD surface plot for time-varying variables in MC simulations
0.05
0
(N=1000; T=5, xi=1)
(N=1000; T=10, xi=2)
(N=1000; T=10, xi=1)
(N=1000; T=10, xi=0.5)
(N=1000; T=5, xi=2)
T
VD
FE
1
IC
A
1
TIC
H
Q
H
1
TH
IC
B
T- tat
s
C
TH
H
H
38
(N=1000; T=5, xi=0.5)
(N=500; T=10, xi=2)
(N=500; T=10, xi=1)
(N=500; 10=5, xi=0.5)
(N=500; T=5, xi=2)
(N=500; T=5, xi=1)
(N=500; T=5, xi=0.5)
(N=100; T=10, xi=2)
(N=100; T=10, xi=1)
(N=100; T=10, xi=0.5)
(N=100; T=5, xi=2)
(N=100; T=5, xi=1)
(N=100; T=5, xi=0.5)
Figure 8: NORMSQD surface plot for time-varying variables in MC simulations
0.1
0
(N=1000; T=10, xi=2)
(N=1000; T=10, xi=1)
(N=1000; T=10, xi=0.5)
(N=1000; T=5, xi=2)
VD
FE
T
1
IC
A
1
TIC
H
Q
H
1
TH
IC
B
Tat
st
C
TH
H
H
39
(N=1000; T=5, xi=1)
(N=1000; T=5, xi=0.5)
(N=500; T=10, xi=2)
(N=500; T=10, xi=1)
(N=500; 10=5, xi=0.5)
(N=500; T=5, xi=2)
(N=500; T=5, xi=1)
(N=500; T=5, xi=0.5)
(N=100; T=10, xi=2)
(N=100; T=10, xi=1)
(N=100; T=10, xi=0.5)
(N=100; T=5, xi=2)
(N=100; T=5, xi=1)
(N=100; T=5, xi=0.5)
Figure 9: NOMAD surface plot for time-fixed variables in MC simulations
1
0
(N=1000; T=10, xi=2)
(N=1000; T=10, xi=1)
(N=1000; T=10, xi=0.5)
(N=1000; T=5, xi=2)
H
VD
FE
T
1
IC
A
1
TIC
H
Q
H
1
TH
IC
B
Tat
st
C
TH
H
40
(N=1000; T=5, xi=1)
(N=1000; T=5, xi=0.5)
(N=500; T=10, xi=2)
(N=500; T=10, xi=1)
(N=500; 10=5, xi=0.5)
(N=500; T=5, xi=2)
(N=500; T=5, xi=1)
(N=500; T=5, xi=0.5)
(N=100; T=10, xi=2)
(N=100; T=10, xi=1)
(N=100; T=10, xi=0.5)
(N=100; T=5, xi=2)
(N=100; T=5, xi=1)
(N=100; T=5, xi=0.5)
Figure 10: NORMSQD surface plot for time-fixed variables in MC simulations
1
0
Table 1: NOMAD and NORMSQD averaged over all MC simulations
Crit.
NOMAD
NORMSQD
Time-varying
FEVD
HT
HT-Cstat
HT-BIC1
HT-HQIC1
HT-AIC1
0.0009
0.0029
0.0030
0.0086
0.0099
0.0058
0.0321
0.0292
0.0321
0.0337
0.0346
0.0329
Time-fixed
FEVD
HT
HT-Cstat
HT-BIC1
HT-HQIC1
HT-AIC1
0.4105
0.1911
0.6009
0.6171
0.6238
0.6231
0.0672
0.1888
0.2615
0.1990
0.1952
0.2132
All variables
FEVD
HT
HT-Cstat
HT-BIC1
HT-HQIC1
HT-AIC1
0.2057
0.0970
0.3019
0.3129
0.3168
0.3144
0.0497
0.1090
0.1468
0.1164
0.1149
0.1230
Note: For details about the Monte Carlo simulation setup see text.
41
Figure 11: NOMAD and NORMSQD values averaged over all MC simulations
0.35
NOMAD
NORMSQD
0.3
0.25
0.2
0.15
0.1
0.05
0
FEVD
HT
HT-Cstat
42
HT-BIC1
HT-HQIC1
HT-AIC1
Table 2: Gravity model for EU wide Export flows for German states (NUTS1 level)
Log(EX)
Log(GDPi )
Log(GDPj )
Log(P OPi )
Log(P OPj )
Log(P RODi )
Log(DISTij )
SIM
RLF
EMU
EAST
BORDER
CEEC
No. of obs.
No. of Groups
Time effects
Wald test (P-val.)
P-value of BP LM
(POLS/REM)
P-value of F-Test
(POLS/FEM)
Hausman m-stat.
(REM/FEM)
DWH endogeneity test
(P-value)
Sargan overid. test
(P-value)
C-Statistic for Distij
(P-value)
Pagan-Hall IV het.test
(P-value)
POLS
1,04∗∗∗
(0,135)
0,64∗∗∗
(0,026)
0,03
(0,132)
0,19∗∗∗
(0,025)
-0,15
(0,241)
-0,87∗∗∗
(0,021)
-0,03∗∗∗
(0,011)
0,01
(0,011)
0,45∗∗∗
(0,029)
-0,80∗∗∗
(0,039)
0,28∗∗∗
(0,050)
0,47∗∗∗
(0,055)
4784
REM
0,35∗
(0,034)
0,31∗∗∗
(0,034)
0.69∗∗∗
(0,197)
0,48∗∗∗
(0,041)
2,11∗∗∗
(0,228)
-1,04∗∗∗
(0,052)
-0,17∗∗∗
(0,052)
0,03∗∗∗
(0,008)
0,36∗∗∗
(0,019)
-0,38∗∗∗
(0,075)
0,26∗
(0,150)
-0,20∗∗
(0,086)
4784
368
yes
(0.00)
0.00
FEM
0,83∗∗
(0,273)
0,34∗∗∗
(0,044)
-1,38∗∗∗
(0,398)
1,79∗∗∗
(0,302)
1,48∗∗∗
(0,275)
(dropped)
-0,18∗∗∗
(0,062)
0,03∗∗∗
(0,008)
0,31∗∗∗
(0,021)
(dropped)
(dropped)
(dropped)
4784
368
yes
(0.00)
FEVD
0,83∗∗∗
(0,273)#
0,34∗∗
(0,044)#
-1,38∗∗
(0,398)#
1,79∗∗∗
(0,302)#
1,48∗∗∗
(0,275)#
-0,97∗∗∗
(0,021)#
-0,18∗∗∗
(0,048)#
0,03
(0,044)#
0,31∗∗∗
(0,054)#
-1,03∗∗∗
(0,043)#
0,07∗∗∗
(0,008)#
0,93∗∗∗
(0,063)#
4784
368
yes
(0.00)
HT$
0,87∗∗∗
(0,271)
0,35∗∗∗
(0,043)
0,18
(0,263)
0,38∗∗∗
(0,084)
1,76∗∗∗
(0,268)
-1,73∗∗∗
(0,403)
-0,29∗∗∗
(0,039)
0,03∗∗∗
(0,007)
0,34∗∗∗
(0,019)
-0,26∗∗
(0,110)
-0,38
(0,438)
-0,22∗
(0,131)
4784
368
yes
(0.00)
0.00
147.2
(0.00)
25.14
(0.00)
6.25
(0.05)
14.12
(0.00)
35.9
(0.10)
Note: ***, **, * = denote significance levels at the 1%, 5% and 10% level respectively. Standard errors are
robust to heteroskedasticity, # = corrected SEs for the FEVD estimator based on the xtfevd Stata routine
provided by Plümper & Tröger (2007). $ = Using the C-Statistic based downward testing algorithm with group
means of X1 = [GDPj,t , P OPi,t , RLFij,t ] as IVs for Z2 = [DISTij ].
43
44
0.000
-0.011
-0.028
-0.025
-0.032
-0.024
-0.024
-0.015
-0.016
-0.001
-0.009
-0.006
-0.006
-0.007
-0.005
-0.005
-0.003
-0.003
-0.001
0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
0.020
0.016
0.019
0.019
0.020
0.019
0.019
0.019
0.019
0.029
0.022
0.031
0.031
0.032
0.030
0.030
0.030
0.030
0.091
0.073
0.065
0.059
0.061
0.077
0.077
0.085
0.085
β11
s.d.
0.020
0.016
0.019
0.019
0.020
0.019
0.019
0.019
0.019
0.029
0.024
0.032
0.031
0.032
0.031
0.031
0.030
0.030
0.091
0.073
0.070
0.064
0.068
0.081
0.081
0.086
0.086
rmse
-0.001
-0.003
-0.002
-0.002
-0.002
-0.002
-0.002
-0.001
-0.001
-0.002
0.006
-0.002
-0.001
-0.003
-0.001
-0.001
-0.002
-0.002
-0.005
0.007
-0.015
-0.012
-0.017
-0.009
-0.009
-0.006
-0.006
bias
0.022
0.015
0.021
0.021
0.021
0.021
0.021
0.022
0.022
0.031
0.023
0.032
0.032
0.033
0.031
0.031
0.030
0.030
0.085
0.065
0.062
0.057
0.058
0.071
0.071
0.078
0.078
β12
s.d.
0.022
0.016
0.021
0.021
0.021
0.021
0.021
0.022
0.022
0.031
0.023
0.032
0.032
0.033
0.031
0.031
0.030
0.030
0.085
0.065
0.064
0.058
0.061
0.072
0.072
0.078
0.078
rmse
Note: For details about the Monte Carlo simulation setup see text.
bias
Coef.
Crit.
-0.002
-0.002
0.004
0.004
0.005
0.004
0.004
0.001
0.001
-0.002
-0.001
-0.006
-0.007
-0.006
-0.006
-0.006
-0.004
-0.004
0.001
0.002
0.066
0.076
0.070
0.043
0.043
0.023
0.023
bias
β21
s.d.
rmse
N = 100
0.093 0.093
0.093 0.093
0.096 0.116
0.095 0.122
0.096 0.119
0.096 0.105
0.096 0.105
0.094 0.097
0.094 0.097
N = 500
0.031 0.031
0.031 0.031
0.031 0.032
0.030 0.031
0.033 0.033
0.031 0.031
0.031 0.031
0.031 0.031
0.031 0.031
N = 1000
0.021 0.021
0.021 0.021
0.020 0.020
0.020 0.020
0.019 0.020
0.020 0.020
0.020 0.020
0.021 0.021
0.021 0.021
0.000
0.000
-0.005
-0.005
-0.006
-0.005
-0.005
-0.003
-0.003
-0.003
-0.002
0.011
0.012
0.013
0.009
0.009
0.003
0.003
-0.001
-0.001
0.052
0.067
0.055
0.029
0.029
0.009
0.009
bias
0.021
0.021
0.020
0.020
0.020
0.021
0.021
0.021
0.021
0.029
0.029
0.029
0.029
0.029
0.029
0.029
0.030
0.030
0.090
0.090
0.108
0.106
0.108
0.106
0.106
0.095
0.095
β22
s.d.
0.021
0.021
0.021
0.020
0.020
0.021
0.021
0.021
0.021
0.029
0.029
0.031
0.031
0.032
0.030
0.031
0.030
0.030
0.090
0.090
0.120
0.125
0.121
0.110
0.110
0.095
0.095
rmse
-0.252
0.001
-0.451
-0.457
-0.455
-0.440
-0.440
-0.410
-0.410
-0.249
-0.034
-0.426
-0.439
-0.428
-0.420
-0.420
-0.390
-0.390
-0.188
0.095
-0.229
-0.185
-0.211
-0.313
-0.313
-0.381
-0.381
bias
0.033
0.039
0.077
0.062
0.068
0.102
0.102
0.137
0.137
0.047
0.037
0.106
0.090
0.108
0.113
0.113
0.140
0.140
0.133
0.412
0.293
0.279
0.295
0.310
0.310
0.298
0.298
γ12
s.d.
Table A.1: Monte Carlo simulation results with N = (100, 500, 1000) for T = 5 and ξ = 1
0.254
0.039
0.458
0.461
0.460
0.451
0.451
0.432
0.432
0.254
0.050
0.439
0.448
0.441
0.435
0.435
0.414
0.414
0.230
0.422
0.372
0.335
0.362
0.440
0.440
0.484
0.484
rmse
0.591
0.004
0.988
1.001
0.997
0.964
0.964
0.899
0.899
0.582
0.079
0.963
0.989
0.968
0.946
0.946
0.875
0.875
0.425
-0.265
0.590
0.495
0.564
0.740
0.740
0.852
0.852
bias
0.061
0.128
0.141
0.105
0.115
0.202
0.202
0.278
0.278
0.088
0.139
0.202
0.166
0.200
0.231
0.230
0.304
0.304
0.195
1.182
0.668
0.656
0.673
0.653
0.653
0.622
0.622
γ21
s.d.
0.594
0.128
0.998
1.006
1.003
0.985
0.985
0.941
0.941
0.031
0.571
0.521
0.525
0.519
0.525
0.525
0.528
0.528
0.468
1.211
0.891
0.821
0.878
0.987
0.987
1.055
1.055
rmse
45
-0.002
-0.008
-0.007
-0.007
-0.007
-0.004
-0.004
-0.002
-0.002
0.000
-0.002
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.006
0.001
0.002
0.001
0.000
0.000
0.000
0.000
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
0.012
0.009
0.013
0.013
0.013
0.012
0.012
0.012
0.012
0.017
0.015
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.035
0.028
0.040
0.036
0.040
0.038
0.038
0.036
0.036
β11
s.d.
0.012
0.011
0.013
0.013
0.013
0.012
0.012
0.012
0.012
0.017
0.016
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.035
0.029
0.041
0.037
0.041
0.038
0.038
0.036
0.036
rmse
0.001
0.006
0.000
-0.002
-0.001
0.000
0.000
0.000
0.000
0.001
0.003
0.000
0.000
0.000
0.000
0.000
0.001
0.001
-0.001
0.005
-0.005
-0.004
-0.006
-0.002
-0.002
-0.001
-0.001
bias
0.012
0.010
0.013
0.013
0.013
0.013
0.013
0.013
0.013
0.017
0.016
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.038
0.031
0.043
0.038
0.043
0.040
0.040
0.038
0.038
β12
s.d.
0.012
0.012
0.013
0.013
0.013
0.013
0.013
0.013
0.013
0.017
0.016
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.038
0.031
0.043
0.038
0.043
0.040
0.040
0.038
0.038
rmse
Note: For details about the Monte Carlo simulation setup see text.
bias
Coef.
Crit.
-0.001
0.000
-0.001
-0.001
-0.002
-0.001
-0.001
-0.001
-0.001
0.000
0.000
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.012
0.013
0.013
0.007
0.007
0.004
0.004
bias
β21
s.d.
rmse
N = 100
0.040 0.040
0.040 0.040
0.042 0.044
0.042 0.044
0.043 0.045
0.040 0.040
0.040 0.040
0.040 0.040
0.040 0.040
N = 500
0.018 0.018
0.018 0.018
0.018 0.018
0.018 0.018
0.018 0.018
0.018 0.018
0.018 0.018
0.018 0.018
0.018 0.018
N = 1000
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.001
0.001
0.002
0.002
0.003
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.004
0.002
0.005
0.001
0.001
0.000
0.000
bias
0.012
0.012
0.013
0.013
0.013
0.013
0.013
0.012
0.012
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.037
0.037
0.041
0.040
0.041
0.039
0.039
0.038
0.038
β22
s.d.
0.012
0.012
0.013
0.013
0.013
0.013
0.013
0.012
0.012
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.037
0.037
0.041
0.040
0.042
0.039
0.039
0.038
0.038
rmse
-0.262
0.001
-0.340
-0.489
-0.410
-0.312
-0.312
-0.280
-0.280
-0.257
0.362
-0.436
-0.436
-0.436
-0.437
-0.437
-0.407
-0.407
-0.314
-0.177
-0.435
-0.433
-0.432
-0.447
-0.447
-0.452
-0.452
bias
0.019
0.019
0.192
0.042
0.161
0.192
0.192
0.188
0.188
0.029
0.064
0.052
0.051
0.051
0.053
0.053
0.159
0.159
0.069
0.056
0.146
0.137
0.150
0.133
0.133
0.140
0.140
γ12
s.d.
Table A.2: Monte Carlo simulation results with N = (100, 500, 1000) for T = 10 and ξ = 1
0.263
0.019
0.390
0.491
0.440
0.366
0.366
0.337
0.337
0.258
0.367
0.439
0.439
0.439
0.440
0.440
0.437
0.437
0.321
0.185
0.459
0.454
0.457
0.467
0.467
0.474
0.474
rmse
0.595
0.027
0.697
0.998
0.837
0.642
0.642
0.577
0.577
0.589
-0.898
0.945
0.945
0.945
0.945
0.945
0.872
0.872
0.630
0.304
0.825
0.864
0.822
0.837
0.837
0.842
0.842
bias
0.035
0.056
0.376
0.060
0.316
0.376
0.376
0.368
0.368
0.051
0.181
0.085
0.083
0.083
0.085
0.085
0.355
0.355
0.111
0.162
0.193
0.195
0.200
0.191
0.191
0.200
0.200
γ21
s.d.
0.596
0.062
0.792
1.000
0.895
0.744
0.744
0.685
0.685
0.591
0.916
0.949
0.949
0.949
0.948
0.948
0.941
0.941
0.640
0.344
0.848
0.886
0.846
0.858
0.858
0.865
0.865
rmse
46
0.000
-0.012
-0.025
-0.027
-0.026
-0.019
-0.019
-0.011
-0.011
-0.001
-0.007
-0.004
-0.004
-0.004
-0.003
-0.003
-0.002
-0.002
-0.001
0.001
0.000
0.000
0.000
0.000
0.000
-0.001
-0.001
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
0.013
0.011
0.013
0.013
0.013
0.013
0.013
0.013
0.013
0.020
0.016
0.022
0.022
0.022
0.022
0.022
0.020
0.020
0.045
0.041
0.039
0.035
0.039
0.045
0.045
0.046
0.046
β11
s.d.
0.013
0.011
0.013
0.013
0.013
0.013
0.013
0.013
0.013
0.020
0.017
0.022
0.022
0.022
0.022
0.022
0.020
0.020
0.045
0.043
0.047
0.045
0.047
0.049
0.049
0.048
0.048
rmse
0.000
-0.003
-0.002
-0.002
-0.002
-0.002
-0.002
-0.001
-0.001
-0.001
0.005
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.002
0.004
-0.008
-0.008
-0.008
-0.004
-0.004
0.000
0.000
bias
0.015
0.012
0.014
0.014
0.014
0.014
0.014
0.015
0.015
0.021
0.016
0.023
0.023
0.023
0.023
0.023
0.021
0.021
0.043
0.038
0.034
0.032
0.034
0.039
0.039
0.041
0.041
β12
s.d.
0.015
0.012
0.015
0.015
0.015
0.014
0.014
0.015
0.015
0.021
0.017
0.023
0.023
0.023
0.023
0.023
0.021
0.021
0.043
0.038
0.035
0.033
0.035
0.039
0.039
0.041
0.041
rmse
Note: For details about the Monte Carlo simulation setup see text.
bias
Coef.
Crit.
-0.001
-0.001
0.002
0.002
0.003
0.002
0.002
0.000
0.000
-0.001
-0.001
-0.004
-0.004
-0.005
-0.004
-0.004
-0.002
-0.002
0.000
0.002
0.018
0.020
0.019
0.014
0.014
0.009
0.009
bias
β21
s.d.
rmse
N = 100
0.047 0.047
0.047 0.047
0.047 0.051
0.050 0.053
0.048 0.052
0.046 0.048
0.046 0.048
0.046 0.047
0.046 0.047
N = 500
0.020 0.020
0.020 0.020
0.020 0.020
0.020 0.020
0.020 0.020
0.020 0.021
0.020 0.021
0.021 0.021
0.021 0.021
N = 1000
0.014 0.014
0.014 0.014
0.013 0.013
0.013 0.013
0.013 0.013
0.014 0.014
0.014 0.014
0.014 0.014
0.014 0.014
0.000
0.000
-0.003
-0.003
-0.003
-0.003
-0.003
-0.001
-0.001
-0.002
-0.001
0.007
0.007
0.007
0.005
0.005
0.001
0.001
-0.001
0.001
0.001
0.002
0.001
-0.001
-0.001
-0.002
-0.002
bias
0.014
0.014
0.013
0.013
0.013
0.014
0.014
0.014
0.014
0.019
0.019
0.019
0.019
0.019
0.020
0.020
0.021
0.021
0.045
0.045
0.047
0.049
0.048
0.044
0.044
0.045
0.045
β22
s.d.
0.014
0.014
0.014
0.014
0.014
0.014
0.014
0.014
0.014
0.019
0.019
0.020
0.020
0.020
0.021
0.021
0.021
0.021
0.045
0.045
0.047
0.049
0.048
0.044
0.044
0.045
0.045
rmse
-0.307
0.002
-0.463
-0.466
-0.466
-0.449
-0.449
-0.392
-0.392
-0.307
-0.047
-0.444
-0.451
-0.448
-0.430
-0.430
-0.382
-0.382
-0.336
-0.050
-0.389
-0.382
-0.387
-0.409
-0.409
-0.424
-0.424
bias
0.025
0.026
0.057
0.043
0.043
0.095
0.095
0.164
0.164
0.036
0.025
0.071
0.061
0.064
0.099
0.099
0.146
0.146
0.089
0.250
0.124
0.127
0.128
0.125
0.125
0.148
0.148
γ12
s.d.
Table A.3: Monte Carlo simulation results with N = (100, 500, 1000) for T = 5 and ξ = 2
0.308
0.026
0.466
0.468
0.468
0.459
0.459
0.425
0.425
0.309
0.053
0.450
0.455
0.453
0.441
0.441
0.409
0.409
0.347
0.255
0.409
0.402
0.408
0.428
0.428
0.449
0.449
rmse
0.720
0.001
0.995
1.002
1.002
0.966
0.966
0.846
0.846
0.714
0.106
0.982
0.997
0.991
0.948
0.948
0.837
0.837
0.769
0.053
0.959
0.954
0.955
0.974
0.974
0.975
0.975
bias
0.049
0.085
0.104
0.069
0.069
0.192
0.192
0.345
0.345
0.071
0.092
0.134
0.103
0.112
0.204
0.204
0.311
0.311
0.164
0.698
0.263
0.282
0.275
0.248
0.248
0.315
0.315
γ21
s.d.
0.722
0.085
1.000
1.004
1.004
0.985
0.985
0.914
0.914
0.718
0.140
0.991
1.002
0.997
0.970
0.970
0.893
0.893
0.786
0.699
0.995
0.994
0.994
1.005
1.005
1.024
1.024
rmse
47
-0.001
-0.005
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
0.000
-0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.000
-0.004
0.001
0.003
0.001
0.000
0.000
0.000
0.000
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
0.008
0.007
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.011
0.011
0.012
0.012
0.012
0.012
0.012
0.012
0.012
0.023
0.021
0.023
0.021
0.023
0.022
0.022
0.022
0.022
β11
s.d.
0.008
0.008
0.008
0.009
0.009
0.008
0.008
0.008
0.008
0.011
0.011
0.012
0.012
0.012
0.012
0.012
0.012
0.012
0.023
0.021
0.023
0.021
0.023
0.022
0.022
0.022
0.022
rmse
0.000
0.003
0.000
-0.002
-0.001
0.000
0.000
0.000
0.000
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.001
0.003
0.000
0.001
0.000
0.000
0.000
0.000
0.000
bias
0.008
0.008
0.009
0.008
0.009
0.009
0.009
0.009
0.009
0.011
0.011
0.012
0.012
0.012
0.012
0.012
0.012
0.012
0.025
0.022
0.024
0.023
0.024
0.024
0.024
0.024
0.024
β12
s.d.
0.008
0.008
0.009
0.009
0.009
0.009
0.009
0.009
0.009
0.011
0.011
0.012
0.012
0.012
0.012
0.012
0.012
0.012
0.025
0.022
0.024
0.023
0.024
0.024
0.024
0.024
0.024
rmse
Note: For details about the Monte Carlo simulation setup see text.
bias
Coef.
Crit.
0.000
0.000
0.000
0.000
-0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.004
0.004
0.004
0.003
0.003
0.001
0.001
bias
β21
s.d.
rmse
N = 100
0.027 0.027
0.027 0.027
0.026 0.026
0.026 0.026
0.026 0.026
0.027 0.027
0.027 0.027
0.027 0.027
0.027 0.027
N = 500
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
0.012 0.012
N = 1000
0.008 0.008
0.008 0.008
0.008 0.008
0.008 0.008
0.008 0.008
0.008 0.008
0.008 0.008
0.008 0.008
0.008 0.008
0.000
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
bias
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.012
0.012
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.025
0.025
0.025
0.025
0.025
0.025
0.025
0.025
0.025
β22
s.d.
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.008
0.012
0.012
0.011
0.011
0.011
0.011
0.011
0.011
0.011
0.024
0.025
0.025
0.025
0.025
0.025
0.025
0.025
0.025
rmse
-0.321
-0.011
-0.314
-0.498
-0.352
-0.307
-0.307
-0.303
-0.303
-0.308
0.637
-0.446
-0.446
-0.446
-0.443
-0.443
-0.401
-0.401
-0.366
-0.218
-0.485
-0.484
-0.485
-0.480
-0.480
-0.464
-0.464
bias
0.015
0.014
0.188
0.029
0.186
0.185
0.185
0.184
0.184
0.021
0.073
0.038
0.038
0.038
0.075
0.075
0.224
0.224
0.051
0.038
0.082
0.072
0.079
0.094
0.094
0.109
0.109
γ12
s.d.
Table A.4: Monte Carlo simulation results with N = (100, 500, 1000) for T = 10 and ξ = 2
0.321
0.018
0.366
0.499
0.398
0.358
0.358
0.355
0.355
0.309
0.641
0.448
0.448
0.448
0.449
0.449
0.459
0.459
0.369
0.222
0.491
0.489
0.491
0.489
0.489
0.476
0.476
rmse
0.724
0.061
0.638
1.002
0.715
0.625
0.625
0.618
0.618
0.711
-1.582
0.959
0.959
0.959
0.950
0.950
0.850
0.850
0.743
0.376
0.916
0.928
0.916
0.903
0.903
0.872
0.872
bias
0.028
0.040
0.361
0.039
0.360
0.355
0.355
0.352
0.352
0.041
0.194
0.055
0.055
0.055
0.160
0.160
0.520
0.520
0.086
0.105
0.120
0.109
0.117
0.142
0.142
0.173
0.173
γ21
s.d.
0.725
0.073
0.733
1.002
0.800
0.718
0.718
0.711
0.711
0.712
1.593
0.960
0.960
0.960
0.963
0.963
0.996
0.996
0.747
0.391
0.924
0.934
0.924
0.914
0.914
0.889
0.889
rmse
48
0.000
-0.011
-0.028
-0.025
-0.032
-0.024
-0.024
-0.015
-0.016
-0.001
-0.008
-0.012
-0.009
-0.014
-0.008
-0.008
-0.005
-0.005
-0.002
0.000
-0.004
-0.002
-0.005
-0.003
-0.003
-0.002
-0.002
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
0.026
0.021
0.028
0.027
0.029
0.026
0.026
0.026
0.026
0.039
0.029
0.042
0.041
0.040
0.040
0.040
0.039
0.039
0.091
0.073
0.065
0.059
0.061
0.077
0.077
0.085
0.085
β11
s.d.
0.026
0.021
0.029
0.027
0.030
0.026
0.026
0.026
0.026
0.039
0.030
0.043
0.042
0.043
0.041
0.041
0.039
0.039
0.091
0.073
0.070
0.064
0.068
0.081
0.081
0.086
0.086
rmse
-0.001
-0.003
-0.005
-0.004
-0.006
-0.003
-0.003
-0.003
-0.003
-0.002
0.004
-0.009
-0.007
-0.011
-0.005
-0.005
-0.003
-0.003
-0.005
0.007
-0.015
-0.012
-0.017
-0.009
-0.009
-0.006
-0.006
bias
0.029
0.020
0.029
0.028
0.030
0.028
0.028
0.029
0.029
0.041
0.029
0.042
0.042
0.041
0.040
0.040
0.039
0.039
0.085
0.065
0.062
0.057
0.058
0.071
0.071
0.078
0.078
β12
s.d.
0.029
0.020
0.030
0.028
0.031
0.028
0.028
0.029
0.029
0.041
0.029
0.043
0.042
0.042
0.040
0.040
0.039
0.039
0.085
0.065
0.064
0.058
0.061
0.072
0.072
0.078
0.078
rmse
Note: For details about the Monte Carlo simulation setup see text.
bias
Coef.
Crit.
-0.002
-0.002
0.006
0.007
0.008
0.004
0.004
0.002
0.002
-0.002
-0.002
-0.001
-0.003
0.013
-0.005
-0.005
-0.005
-0.005
0.001
0.002
0.066
0.076
0.070
0.043
0.043
0.023
0.023
bias
β21
s.d.
rmse
N = 100
0.093 0.093
0.093 0.093
0.096 0.116
0.095 0.122
0.096 0.119
0.096 0.105
0.096 0.105
0.094 0.097
0.094 0.097
N = 500
0.041 0.041
0.041 0.041
0.048 0.048
0.048 0.048
0.065 0.066
0.045 0.045
0.045 0.045
0.042 0.042
0.042 0.042
N = 1000
0.028 0.028
0.028 0.028
0.028 0.029
0.028 0.029
0.029 0.030
0.026 0.027
0.026 0.027
0.027 0.027
0.027 0.027
0.000
0.000
-0.004
-0.006
-0.003
-0.005
-0.005
-0.003
-0.003
-0.003
-0.003
0.018
0.020
0.029
0.012
0.012
0.005
0.005
-0.001
-0.001
0.052
0.067
0.055
0.029
0.029
0.009
0.009
bias
0.028
0.028
0.029
0.028
0.030
0.028
0.028
0.028
0.028
0.039
0.039
0.044
0.043
0.056
0.041
0.041
0.039
0.039
0.090
0.090
0.108
0.106
0.108
0.106
0.106
0.095
0.095
β22
s.d.
0.028
0.028
0.029
0.028
0.031
0.028
0.028
0.029
0.029
0.039
0.039
0.047
0.047
0.063
0.043
0.043
0.040
0.040
0.090
0.090
0.120
0.125
0.121
0.110
0.110
0.095
0.095
rmse
-0.165
-0.002
-0.410
-0.442
-0.409
-0.408
-0.408
-0.398
-0.398
-0.161
-0.023
-0.363
-0.405
-0.327
-0.376
-0.376
-0.370
-0.370
-0.188
0.095
-0.229
-0.185
-0.211
-0.313
-0.313
-0.381
-0.381
bias
0.036
0.051
0.141
0.097
0.142
0.144
0.144
0.156
0.156
0.051
0.051
0.182
0.158
0.213
0.174
0.174
0.173
0.173
0.133
0.412
0.293
0.279
0.295
0.310
0.310
0.298
0.298
γ12
s.d.
Table A.5: Monte Carlo simulation results with N = (100, 500, 1000) for T = 5 and ξ = 0.5
0.169
0.051
0.433
0.453
0.433
0.433
0.433
0.427
0.427
0.169
0.056
0.406
0.435
0.390
0.414
0.414
0.409
0.409
0.230
0.422
0.372
0.335
0.362
0.440
0.440
0.484
0.484
rmse
0.390
0.010
0.919
0.985
0.918
0.911
0.911
0.881
0.881
0.378
0.057
0.851
0.931
0.775
0.866
0.866
0.845
0.845
0.425
-0.265
0.590
0.495
0.564
0.740
0.740
0.852
0.852
bias
0.056
0.170
0.251
0.168
0.251
0.269
0.269
0.302
0.302
0.080
0.188
0.344
0.303
0.427
0.337
0.337
0.360
0.360
0.195
1.182
0.668
0.656
0.673
0.653
0.653
0.622
0.622
γ21
s.d.
0.394
0.171
0.952
1.000
0.952
0.949
0.949
0.931
0.931
0.386
0.196
0.918
0.979
0.885
0.929
0.929
0.918
0.918
0.468
1.211
0.891
0.821
0.878
0.987
0.987
1.055
1.055
rmse
49
-0.003
-0.008
-0.016
-0.016
-0.016
-0.008
-0.008
-0.004
-0.004
0.000
-0.003
0.001
0.001
0.001
0.001
0.001
0.001
0.001
0.000
-0.006
0.001
0.002
0.001
0.000
0.000
0.000
0.000
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
FEVD
HT
HT-BIC1
HT-Cstat
HT-BIC2
HT-HQIC1
HT-HQIC2
HT-AIC1
HT-AIC2
0.016
0.011
0.016
0.017
0.017
0.016
0.016
0.016
0.016
0.023
0.017
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.047
0.036
0.050
0.047
0.048
0.050
0.050
0.049
0.049
β11
s.d.
0.016
0.012
0.016
0.017
0.017
0.016
0.016
0.016
0.016
0.023
0.017
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.047
0.037
0.053
0.049
0.051
0.051
0.051
0.049
0.049
rmse
0.001
0.006
0.000
-0.001
-0.001
0.000
0.000
0.001
0.001
0.001
0.004
0.000
0.000
0.000
0.000
0.000
0.001
0.001
-0.001
0.004
-0.016
-0.015
-0.015
-0.007
-0.007
-0.003
-0.003
bias
0.017
0.013
0.017
0.017
0.017
0.017
0.017
0.016
0.016
0.023
0.018
0.023
0.023
0.023
0.023
0.023
0.023
0.023
0.051
0.039
0.053
0.050
0.050
0.052
0.052
0.051
0.051
β12
s.d.
0.017
0.014
0.017
0.017
0.017
0.017
0.017
0.016
0.016
0.023
0.019
0.023
0.022
0.023
0.023
0.023
0.023
0.023
0.051
0.039
0.055
0.052
0.053
0.053
0.053
0.051
0.051
rmse
Note: For details about the Monte Carlo simulation setup see text.
bias
Coef.
Crit.
-0.001
0.000
-0.002
-0.002
-0.003
-0.002
-0.002
-0.001
-0.001
0.000
0.000
0.001
0.001
0.001
0.001
0.001
0.000
0.000
0.000
0.000
0.026
0.031
0.035
0.013
0.013
0.005
0.005
bias
β21
s.d.
rmse
N = 100
0.053 0.053
0.053 0.053
0.063 0.068
0.061 0.069
0.070 0.078
0.058 0.059
0.058 0.059
0.052 0.053
0.052 0.053
N = 500
0.025 0.025
0.025 0.025
0.023 0.023
0.023 0.023
0.023 0.023
0.023 0.023
0.023 0.023
0.024 0.024
0.024 0.024
N = 1000
0.015 0.015
0.015 0.015
0.016 0.016
0.016 0.016
0.015 0.016
0.016 0.016
0.016 0.016
0.016 0.016
0.016 0.016
0.001
0.001
0.004
0.004
0.007
0.003
0.003
0.001
0.001
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.017
0.016
0.026
0.007
0.007
0.002
0.002
bias
0.016
0.016
0.017
0.017
0.017
0.017
0.017
0.017
0.017
0.023
0.023
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.049
0.049
0.059
0.060
0.068
0.055
0.055
0.051
0.051
β22
s.d.
0.016
0.016
0.018
0.018
0.018
0.017
0.017
0.017
0.017
0.023
0.023
0.022
0.022
0.022
0.022
0.022
0.022
0.022
0.049
0.049
0.061
0.062
0.073
0.056
0.056
0.051
0.051
rmse
-0.172
0.005
-0.389
-0.479
-0.445
-0.349
-0.349
-0.293
-0.293
-0.176
0.194
-0.420
-0.420
-0.420
-0.420
-0.420
-0.409
-0.409
-0.226
-0.129
-0.335
-0.337
-0.304
-0.375
-0.375
-0.392
-0.392
bias
0.021
0.024
0.163
0.055
0.113
0.182
0.182
0.188
0.188
0.031
0.055
0.067
0.067
0.067
0.066
0.066
0.115
0.115
0.078
0.076
0.202
0.194
0.221
0.188
0.188
0.176
0.176
γ12
s.d.
Table A.6: Monte Carlo simulation results with N = (100, 500, 1000) for T = 10 and ξ = 0.5
0.173
0.024
0.421
0.482
0.459
0.393
0.393
0.348
0.348
0.179
0.201
0.426
0.425
0.425
0.425
0.425
0.424
0.424
0.239
0.149
0.391
0.389
0.376
0.419
0.419
0.430
0.430
rmse
0.392
0.007
0.807
0.989
0.922
0.725
0.725
0.612
0.612
0.400
-0.482
0.919
0.919
0.919
0.918
0.918
0.877
0.877
0.446
0.222
0.668
0.710
0.616
0.703
0.703
0.719
0.719
bias
0.033
0.073
0.323
0.083
0.217
0.364
0.364
0.377
0.377
0.049
0.169
0.115
0.115
0.115
0.114
0.114
0.223
0.223
0.113
0.229
0.288
0.306
0.345
0.262
0.262
0.246
0.246
γ21
s.d.
0.394
0.073
0.869
0.993
0.947
0.811
0.811
0.719
0.719
0.403
0.511
0.926
0.926
0.926
0.926
0.926
0.905
0.905
0.460
0.319
0.728
0.773
0.706
0.750
0.750
0.759
0.759
rmse
Table A.7: Data description and source for Export model
Variable
EXijt
Description
Export volume, nominal values, in Mio.
GDPit
Gross Domestic Product, nominal values, in Mio.
GDPjt
POPit
Gross Domestic Product, nominal values, in Mio.
Population, in 1000
POPjt
Population, in 1000
SIMijt
RLFijt
2 2 GDPjt
it
SIM = log 1 − GDPGDP
−
+GDP
GDP
+GDP
it
jt
it
jt
GDPjt it
−
RLF = log GDP
P OPit
P OPjt
EMPit
Employment, in 1000
EMPjt
Employment, in 1000
PRODit
P rodit =
PRODjt
P rodjt =
DISTij
Distance between state capital for Germany and national capital for the EU27 countries, in km
EMU
(0,1)-Dummy variable for EMU members since 1999
EAST
(0,1)-Dummy variable for the East German states
CEEC
(0,1)-Dummy variable for the Central and Eastern European countries
(0,1)-Dummy variable for country pairs with a common
border
BORDER
GDPit
EM Pit
GDPjt
EM Pjt
Source
Statistisches Bundesamt
(German statistical office)
VGR der Länder (Statistical
office of the German states)
EUROSTAT
VGR der Länder (Statistical
office of the German states)
Groningen Growth &
Development center (GGDC)
see above
see above
VGR der Länder (Statistical
office of the German states)
AMECO database of the
European Commission
see above
see above
50
Calculation based on
coordinates, obtained from
www.koordinaten.de